Relating insplittings of 2-graphs and of textile systems

TL;DR

This paper explores the relationship between insplitting operations in 2-graphs and textile systems, demonstrating how insplitting in 2-graphs induces conjugacy in associated 2D shift systems and connecting operator algebra with dynamical systems.

Contribution

It establishes a method to reconstruct 2-graph insplitting via textile system insplits, showing that 2-graph insplitting induces conjugacy of dynamical systems, and links algebraic and dynamical perspectives.

Findings

2-graph insplitting induces conjugacy of associated systems

Reconstruction of 2-graph insplitting from textile system insplits

Key role of bottom graph insplit in the relationship

Abstract

The graphical operation of insplitting is key to understanding conjugacy of shifts of finite type (SFTs) in both one and two dimensions. In this paper, we consider two approaches to studying 2-dimensional SFTs: textile systems and rank-2 graphs. Nasu's textile systems describe all two-sided 2D SFTs up to conjugacy, whereas the 2-graphs (higher-rank graphs of rank 2) introduced by Kumjian and Pask yield associated C*-algebras. Both models have a naturally-associated notion of insplitting. We show that these notions do not coincide, raising the question of whether insplitting a 2-graph induces a conjugacy of the associated one-sided 2-dimensional SFTs. Our first main result shows how to reconstruct 2-graph insplitting using textile-system insplits and inversions, and consequently proves that 2-graph insplitting induces a conjugacy of dynamical systems. We also present several other facets of the relationship between 2-graph insplitting and textile-system insplitting. Incorporating an insplit of the bottom graph of the textile system turns out to be key to this relationship. By articulating the connection between operator-algebraic and dynamical notions of insplitting in two dimensions, this article lays the groundwork for a C*-algebraic framework for classifying one-sided conjugacy in higher-dimensional SFTs.